A COMPARATIVE STUDY FOR SOLVING FIRST ORDER LOGISTIC EQUATIONS BY USING STANDARD AND NON-STANDARD FINITE DIFFERENCE METHODS

Abstract

Logistic equations are mostly used to model real life problems. However it is known that in general, these equations do not have analytic solution. For that reason, researchers have tried to solve these problems by means of both numerical methods. Among them, standard finite difference (SFD) is a frequently used method in order to obtain numerical solutions of logistic differential equation for a long time. However there are many mathematical problems for which the SFD models do not perform well. In recent years, nonstandard finite difference (NSFD) method which gets main motivation from SFD method has been applied to various mathematical models for the purpose of getting reliable numerical results.To obtain the best numerical results in which the scheme of the numerical methods that does not allow numerical instabilities to occur for all the step lengths to have a good numerical solution to the logistic differential equation.We considered the logistic differential equation and found the numerical solution using NSFD scheme by applying Mickens rules. The several examples for the comparison we have chosen the same examples and obtained the numerical results by using NSFD and SFD methods. Under SFD methods Euler forward method was relatively more stable than the RK2 and central difference methods and RK2 was more stable than that of the central difference method, then even both methods (NSFD) and (SFD) was in a good agreement, for small step-length , SFD methods exhibits the numerical instability but NSFD discrete models do not exhibit numerical instability. Totally Euler forward method was compared with that of NSFD method and Moreover, the important preliminary concepts, examples, definitions and theorems are present to make the concept of both methods more clear. By comparisons of both numerical methods we obtain NSFD was more accurate totally than that of SFD in solving logistic differential equation.The computations were performed by using MATLAB codes to find numerical results in which the scheme of the numerical methods that does not allow numerical instabilities. MATLAB codes should taken into account for now and the future work. The continuous work on solving logistic equation by using SFD methods, there are a few areas and limited problems that will need to improve in the future work. In order to obtain the best numerical results in which the scheme of the numerical methods that does not allow numerical instabilities to occur for all the step lengths to have a numerical solution of the logistic differential equations